\section{Introduction}
\label{sec:Introduction}
The light transport within volume usually undergoes emission, absorption and scattering through the traversal of volume grids. Such a complicated light propagation produces a number of volumetric shading effects, such as volumetric shadow, multiple scattering, volume caustics and etc, which are crucial for the natural appearance of volume media. Simulation of these volumetric effects presents a challenging rendering problem.

Using full Monte Carlo ray tracing, these volumetric shading effects can be accurately rendered in a general way. However, the prohibitively high computation expense of ray tracing leads itself to be unrealistic. Kajiya and Von Herzen~\cite{Kajiya:1984:RTV} firstly proposed to separate the rendering procedure into two steps. In the first step, the source radiance at each voxel center in the volume is estimated, and then in the second step we march along the view rays to gather the source radiance. Although this method is more efficient than the Monte Carlo ray tracing, due to the dense sampling of source radiance, the radiance estimation of each voxel in the first step still requires substantial computations and can only be done in offline.
The target for achieving real-time rendering performance naturally leads us to the precomputation-based solution.

In recent years, precomputed radiance transfer (PRT) method~\cite{Sloan:2002:PRT} has been extensively investigated in computer graphics community. The motivation of PRT is to render a scene in real-time with complex light propagation being precomputed to save time. Some kind of basis function, such as spherical harmonics, is often adopted to approximately encode large amount of precomputation data. Hence, not only the precomputation data can be stored economically, but also the reconstruction of the light interactions in rendering stage is very efficient. Obviously, it is also possible to apply the PRT technique to render complex volumetric effects. However, usual volume dataset, like $512^3$, contains more than one hundred million voxels, and how to
simulate the light propagation efficiently for such a huge amount of data is non-trivial. 

In this paper, we propose a general precomputation-based volume rendering framework, so-called precomputed volume radiance transfer (PVRT), to achieve real-time rendering of various volumetric effects. Similar as the PRT method, the PVRT framework contains two stages: \emph{precomputation} and \emph{rendering}. In precomputation stage, volumetric photon mapping (VPM) technique~\cite{Jensen:1998:ESL} is incorporated to handle the radiance estimation for each voxel. Different from the standard VPM method, we design a new photon sampling method for volumes under spherical distant lightings to cast photons into volume. Then, instead of gathering photon's energy for each voxel, we present an efficient photon splatting technique for volumetric spherical radiance estimation with low memory consumption.
 
To achieve high-quality all-frequency rendering, both low-frequency spherical harmonics (SH) basis function and high-frequency spherical radial basis function (SRBF) are adopted to encode light transportation.

Volume rendering have been studied extensively in both computer graphics and scientific visualization communities. In computer graphics, the goal is to rendering highly realistic volumetric effects in production quality for animations and movies. Global illumination model is required in most cases but the computational cost could be very high due to the complexity of the physically-based light transportation such as light scattering in inhomogeneous participating media. In scientific visualization, the goal is to render the volume data in a precise manner such that scientists could learn from the visualized result. In addition, interactive visualization further helps understanding the data because it provides the capability to explore the volume in realtime with different conditions, such as view points and lighting environments. Although accurate global illumination could enhance the visualization in human perceptual aspect, it is often neglected when interactivity is more crucial. There are already some works done on interactive volume rendering with global illumination, but the techniques proposed are either based on non-accurate empirical visual approximation or simplified lighting models. In our proposed work, we try to achieve interactive volume rendering with physically-accurate global illumination model. The key to make it work is based on a practical pre-computation strategy, with a additional space and time cost at a reasonable level.
